I How can we define a space?

1. Sep 11, 2017

Tio Barnabe

What are the ways of mathematically defining a space?
We usually define a space by stating some common property shared by the points that make up the space in question, that is, a symmetry of that space. For instance, the ordinary 2-sphere is defined as all points $X$ that are at a distance $c > 0, c \in \mathbb{R}$ from some point $X_0$, the distance being calculated using some specific metric. Does this implicitaly define a coordinate system? It seems that it doesn't, as long as we take the usual definition of a coordinate system as the bijection from a region of the sphere to some other appropriate space through some specific mapping. (This was my first question.)

Now, say we have a real, arbritary object of every-day life; this could be my smartphone. Is there a similar way as above for defining it mathematically? I suppose that if there is, it must be extremely difficult, because the smartphone is a space (or object?) with no patterns... or does it has patterns, but these are so complex that a human brain could not realize them?

Last edited by a moderator: Sep 11, 2017
2. Sep 11, 2017

FactChecker

That depends entirely on what properties the space has -- topological space, Hausdorff space, metric space, vector space, inner product space, etc. There are all kinds of spaces.
This will work if there is a metric on the space to define "distance". Otherwise, this won't work.
If by that you mean a coordinate system that defines a geometry in Euclidean space, then no. You can have distance without any kind of embedding in Euclidean space.
In mathematics, a space of any type has a much more formal definition. It is taking a wrong turn to talk about a smartphone as a space. Any type of mathematical space is defined in terms of its mathematical properties specifically so that mathematical statements can be stated about its elements.

3. Sep 11, 2017

Tio Barnabe

Not necessarily Euclidean Space, we can take again the Sphere as an example. Does your answer change in this case?
Ok. So we can forget a bit about the formal names. Is it possible to mathematically define the smartphone?

4. Sep 11, 2017

FactChecker

I think that the mathematical definition of "distance" allows so much flexibility that it does not force a space to have a shape like you are thinking of.
Not as far as I know. You can mathematically define and analyse some specific properties of a smartphone, but not the entire thing in any one existing mathematical abstraction.

5. Sep 12, 2017

Tio Barnabe

What if we partitionate it in several parts, each of them can then be approximated by a known, easy-to-define, mathematical object. After that, we could say it is the union of all the parts.

6. Sep 12, 2017

FactChecker

I agree (Although "union" would not be the right word. "Product space" is better.). That would end up being a very complicated description. Probably too complicated to do anything with it. Math is intended to be useful and to simplify analysis, so something infinitely complicated is not good math.

7. Sep 12, 2017

WWGD

Usually you can use a Mathematical description if your object has a reasonably clear Mathematical analog, which requires you to specify the type of structure on your phone you are interested in. There are many possible perspectives and types of structure from which we can view and study a phone. Are you interested in any particular perspective or structure? EDIT: Repeating some of what @FactChecker said, The simplest example I can think of is that of the Real number line. We can study it as an ordered field ( Order properties) , as a Metric space, As a group , as a Ring , etc. A slighly less straightforward example would be the use of Topological Data Analysis in which we attach a certain topological space ( usually one in which we can develop a homology theory : CW -Complex, simplicial Complex, etc. ) in a way that the attached space allows for a study of properties of the data set. So ultimately, you need to be more specific on what aspect/structure(s) of the phone you are interested in: is it the way in which the parts are interconnected? Or how phone is connected with other phones through a network, etc.

Good question, BTW

8. Sep 12, 2017

Tio Barnabe

Thanks WWGD

Let's say I want to tell a person who have never seen a phone how it looks like. Furthermore, this person has a knowledge on computer systems, and once I give him/her a description of the phone using analytic geometry, he/she will use this information in a computer, which will form the image of the phone.

In other words, I'm interested in defining the shape of the phone, in the same way we use analytic geometry to define a sphere1, a cylinder etc.

1 - the locus of all points (x, y, z) such that... .

9. Sep 12, 2017

WWGD

Sorry if I am missing something, but, why not just send them a picture of the phone?

10. Sep 12, 2017

Tio Barnabe

That was just an example to explain better what I mean

11. Sep 12, 2017

WWGD

Ah, got it, let me think it through then.

12. Sep 15, 2017

WWGD

Attn : @mathwonk , I think this is a question about parametrization, which you know. about. EDIT: This is the closest Mathematical analog I can think of Tio Bernabe's question: can the (surface of a ) phone be described Mathematically, is equivalent to whether the surface can be parametrized.

Last edited: Sep 15, 2017
13. Dec 14, 2017

wondering12

I have a very simple reference about space. I noticed that most bodies in quantum mechanics, microscopic, star, galaxy, and the entire universe is spherical to begin with. Therefore space is spherical in nature. By induction, mathematical coordinates should be spherical regardless the object under study looks flat or not.

14. Dec 16, 2017

Staff: Mentor

A smartphone is, at heart, a computer, with all of the usual computer components: CPU, memory, input and output devices. These components are not defined mathematically -- they are defined by the circuit and logic diagrams that they are designed from. Below is a block diagram of a generic computer.

Each component in the block diagram can be further described by the discrete logic units that make it up -- transistors, AND gates, OR gates, and so on. Hardware designers typically use software, such as Verilog and others, to design the computer components.

15. Dec 16, 2017

WWGD

I think he's referring to the shape of the phone.

16. Dec 16, 2017

Staff: Mentor

That's not how I interpret his question.
If we're talking about an object that has the shape of a smartphone, then that's a different question, but "mathematically define a smartphone" suggests including the capabilities and features, not just its shape.

17. Dec 16, 2017

WWGD

18. Dec 16, 2017

Staff: Mentor

OK, I stand corrected. I was going by what he wrote in post #1, not what he wrote later.

19. Dec 16, 2017

WWGD

Even then, don't you think the system-wise definition of the phone could be formulated Mathematically? Not the individual components, but the way they relate to other components? This is a good question: what can and does Mathematics describe?

20. Dec 17, 2017

Staff: Mentor

I don't believe so, unless we talk about functions at the software level, pretty much a universal concept in programming. At any rate, I've never seen anything like a mathematical description of a computer system, including what we're talking about here, smartphones.
At a lower level than you're asking about is the design of the hardware itself. Here's some Verilog code whose output would be the digital logic to clear, load, or shift the bits in an 8-bit register. The code entity is a module, which is a kind of function. Of the seven parameters, the first six are inputs to the module, and the last is the output. The hardware that implements this logic would consist of a number of gates and a control unit. BTW, the second comment should say, "Retains value if no control signal is asserted."